Mathematical Description of Topological Gauge Theories

F_p ∼=G. There exists a surjective projection map π:P →M, such that π⁻¹(p) =F_p ∼=G.

The fiberF_pis the orbit of the left (or right) action of the gauge group at a pointp. Fromp= (π(u)) it follows thatF_p ={ug|g ∈G}. The manifold M can be covered with topologically trivial open sets{U_i}. The bundle restricted to a topological trivial subset is trivial and each u=π⁻¹(p) can be described by the coordinates (x∈U_i, g∈G) in the direct productU_i×G.

IfU_i covers the whole manifoldM thenP is trivial and it is local trivial otherwise. There is a further definition of triviality in terms of sections. A local section s_i is a smooth mapping s_i:U_i →π⁻(U_i) such that

(s⁻¹◦π)(p) =id_Fp,

the choice of the local sections s is a choice of a local coordinate system. The transition from one local section to another has the meaning of a gauge transition. Two different local sections sand ˜sare connected by the map g:U →G via the equation

s(p) = ˜s(p)g(p).

Thus the change of a local section corresponds to a gauge transformation and therefore the choice of a local section corresponds to the choice of a local gauge. Two local sets (si, Ui), (sj, Uj) withp∈Ui∩Uj are connected by the equation:

s_j(p) =s_i(p)tⁱ_j(p), tⁱ_j ∈G

however now with the transition function tⁱ_j : Ui ∩Uj → G satisfying the compatibility relations:

t_ii(p) = id_G, p∈U_i (3.3.1)

t_ij(p) = t⁻_ji¹(p), p∈U_i∩U_j (3.3.2) t_ij(p)t_jk(p) = t_ik(p), p∈U_i∩U_j∩U_k. (3.3.3) The last line is also called the cocycle, fixing the cocycle corresponds then to the choice of a gauge. If the bundle is (global) trivial then all transition functions can be chosen to be the identity.

Matter Fields

Formally matter fields are mathematically defined as maps from the bundle P to a vector space V:

φ: P →V (3.3.4)

φ7→φ(ug) =π_V(g⁻¹)φ(u) (3.3.5) with π_V being a representation of the gauge group G in the vector space V. A local repre-sentation we obtain by a local section

φi(x) = (s^∗_iφ)(p) =φ(si(p)).

Parallel Transport, Covariant Derivative and Field Strength

We want to compare the matter fields at different points on our manifold therefore we have to define a parallel transport (connection) on our principal G bundle. So let T_uP be the tangent space at u ∈ P, we divide the tangent space in a so called horizontal and vertical part:

T_uP =V_uP⊕H_uP

with V_uP being tangent to the fibre π⁻¹(π(u)) ∼= G and thus being isomorphic to the Lie algebra Lie(G) of the structure group G. With the mapping u7→ ug, g∈ Gthe horizontal subspaceH_uP is identified with the horizontal subspace atH_ugP. We may define the vertical spaceV_uP as a projection from the tangent spaceT_uP. The projection map is given by a Lie algebra valued connection one formω, which is smooth on all ofP. The horizontal subspace can then be defined by

HuP :={ξ∈TuP|ω(ξ) = 0}.

In physics the gauge fields are called gauge potentials and correspond to the connection one form, usually given in local coordinates. The connection one form ω however is globally de-fined onP. Pulling back the connection one formω with a local sections(chose a coordinate system) gives the local connection one formA_i on each open covering set U_i ⊂M:

A_i :=s^∗_iω,

in this senseA_iis aLie(G) valued one form onU_i×G. Ifsis a global section then the bundle is globally trivial andAbecomes a global connection. With the compatibility relations (3.3.1) we can transform the local connection A_i and A_j on U_i∩U_j by

A_j =t⁻_ij¹A_it_ij+t⁻_ij¹dt_ij. (3.3.6) This is known to be a gauge transformation on the other hand a gauge transformation is also given by the choice of a different section ˜s_i=s_ig, s˜_i, s_i ∈U_i and g∈Gby

A˜_i =g⁻¹A_ig+g⁻¹dg. (3.3.7) The exterior derivative d gives the variation of a field on M, however the variation on the fiber bundle P is given by the covariant derivative

D:=d+ω (3.3.8)

and the curvature form Ω can be obtained by applying the covariant derivative to the con-nection one form:

Ω =Dω=dω+ω∧ω. (3.3.9)

In physics the local curvature is called field strength and is a Lie algebra valued two form, which can be recovered with the pull-back of the curvature

F_i =s^∗_iΩ

with the local section (s_i, U_i) we have

F_i=dA_i+A_i∧A_i and may define the local covariant derivative

D_i =d+A_i. A gauge transformation of the field strength is given by

F_j = t⁻_ij¹F_it_ij, on U_i∩U_j (3.3.10) F˜i = g⁻¹Fig, withs,s˜∈Ui. (3.3.11) We now want to define a mapping Γ from one fiberF_p =π⁻¹(p) to another fiberF_p^′′ =π⁻¹, Γ :F_p →F_p^′′. Let c: [0,1]→M be a smooth path on M with c(0) =p and c(1) =p^′. The pathγ in the principal bundleP is defined byγ =π⁻¹(c) and is called horizontal lift of the path c. The tangent vectors [ ˙γ](u) ∈H_uP are horizontal. The parallel transport is defined as the map

Γ(γ) :π⁻¹(p)→π⁻¹(p^′). (3.3.12) More precisely given a curve c and a fixed u ∈ π⁻¹ there exists only one horizontal lift γ, which fixes then alsou^′ ∈π⁻¹(p^′). Locally the parallel transport is given via the path-integral along the pathc.

u^′ =s_i(p^′)Pe⁻

Rc(1) c(0) Ai

.

The curvature can be recovered from the parallel transport along a closed path c. The difference between initial and end point of the corresponding horizontal lift γ is a measure for the curvature. The horizontal lift is only a closed curve for vanishing curvature thus the parallel transport called Wilson-loop

W_c =Pe^−RcA (3.3.13)

is then zero in this case.

Topology and Fiber Bundles

The construction of a principalGfiber bundle from a given manifoldM and a given structure group G is not unique. Given an open covering {U_i} of the manifold M the fibers π⁻¹(U_i) are glued together corresponding to the choice of the transition functionst_ij. Since the choice of the t_ij is not fixed the bundles are not uniquely defined. This issue is addressed by the classification of bundles by their topology, which characterizes the bundle structure. The task is to divide the bundles into equivalence classes or characteristic classes. The theory of characteristic classes is due to Chern and Weil, they generalized the Gauss-Bonnet theorem to principal bundles.

S

Figure 3.2: A closed manifoldS of genus threeg= 3.

Gauss-Bonnet Theorem

LetS be a closed oriented Riemannian manifold and letK be the Gauss curvature andχ(S) the Euler number of the manifoldS, then the following relation holds:

1 2π

Z

S

K dA=χ(S) (3.3.14)

for example consider the case in figure 3.2 here we have the Euler numberχ(S) = 2−2g=−4 withg being the genus of the oriented closed manifoldS. If the manifold ˜S is only compact and oriented then the formula (3.3.14) changes to

1 2π

Z

S˜

K dA− Z

∂S˜

k_g ds=χ(S) (3.3.15)

which means that the geodesic curvature contributes on the boundary and can be defined for any loop on the surface ∂S˜ see figure 3.3 . This can be viewed as some intrinsic geometric property which we can integrate out and are left with a pure topological number, the Euler number. In the Chern Simons Theory we have some extrinsic geometric property reflected in a compact gauge group Gsay SU(N) which we may integrate out and are then also left with some pure topological properties.

S˜

∂S˜

geodesic curvaturek_gis unique modZ

Figure 3.3: A compact manifold ˜S with boundary ∂S.˜